A006967 Number of graceful permutations of length n.

1, 1, 2, 4, 4, 8, 24, 32, 40, 120, 296, 648, 1328, 3200, 9912, 25592, 55920, 143192, 510696, 1451296, 3497344, 10451824, 38570704, 118914992, 315235872, 1014824752, 3963684496, 13166130152, 37846301904, 130507967088, 533318630936, 1884550215976, 5800121391936

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard

Comments

Also the number of graceful labelings of the path graph P_n. - Eric W. Weisstein, Mar 31 2020

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• H. S. Wilf and N. Yoshimura, Ranking rooted trees and a graceful application, in Discrete Algorithms and Complexity (Proceedings of the Japan-US joint seminar, 1986, Kyoto, Japan), edited by D. Johnson, T. Nishizeki, A. Nozaki and H. S. Wilf, Academic Press, NY, 1987, pp. 341-350.

Links

• Michal Adamaszek, Don Knuth, Table of n, a(n) for n = 0..41, a(41) from Don Knuth.
• M. Adamaszek, Efficient enumeration of graceful permutations, arXiv:math/0608513 [math.CO], 2006.
• Gheorghe Coserea, Solutions for n=5.
• Gheorghe Coserea, Solutions for n=6.
• Gheorghe Coserea, MiniZinc model for generating solutions.
• Don Knuth, This program finds all of the nonisomorphic graceful labelings of the path P_n
• Don Knuth, This program outputs ZDDL for all of the nonisomorphic graceful labelings of the path P_n
• Md Masbaul Alam Polash, M. A. Hakim Newton, Abdul Sattar, Constraint-directed search for all-interval series, Constraints, July 2017, Volume 22, Issue 3, pp 403-431. See page 426.
• Eric Weisstein's World of Mathematics, Graceful Labeling
• Eric Weisstein's World of Mathematics, Graceful Permutation
• Eric Weisstein's World of Mathematics, Path Graph
• J. Wodlinger, Costas arrays, Golomb rulers and wavelength isolation sequence pairs, M.S. Dissertation, Math. Dept., Simon Fraser University, Spring 2012; - From _N. J. A. Sloane_, Oct 15 2012

Crossrefs

Cf. A084894.

Programs

CWEB

@ See Knuth link.

Formula

a(n) = n! - A084894(n). - Jon Perry, Jun 10 2003

Extensions

n=2 term corrected June 1996

a(11)-a(20) from Robert Aldred and Brendan McKay

More terms from Michal Adamaszek (aszek(AT)mimuw.edu.pl), Aug 22 2006

a(0)=1 prepended by Alois P. Heinz, Jan 31 2020

a(41)=1032009647743958000 from Don Knuth, Sep 10 2020